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A note on periods of powers. (English. French summary) Zbl 1360.37097
Summary: Let \(f:X \to X\) be a continuous map defined from a topological space \(X\) into itself. We discuss the problem of analyzing and computing explicitly the set \(\mathrm{Per} (f^p)\) of periods of the \(p\)-th iterate \(f^p\) from the knowledge of the set \(\mathrm{Per} (f)\) of periods of \(f\). In the case of interval or circle maps, that is, \(X=[0,1]\) or \(X=\mathbb S^{1}\), this question was solved in [the author, Int. J. Pure Appl. Math. 72, No. 4, 527–536 (2011; Zbl 1247.37033)]. Now, we present some remarks and advances concerning the set \(\mathrm{Per} (f^p)\) for a continuous interval map, and on the other hand we study and solve the problem when we consider \(\sigma\)-permutation maps, namely, when \(X=[0,1]^k\) for some integer \(k \geq 2\) and the map has the form \(F(x_{1},x_{2}, \dots,x_{k})=(f_{\sigma (1)}(x_{\sigma (1)}),f_{\sigma (2)}(x_{\sigma (2)}), \dots,f_{\sigma (k)}(x_{\sigma (k)}))\), being each \(f_j\) a continuous interval map and \(\sigma\) a cyclic permutation of \(\{1,2,\dots,k\}\). This paper can be seen as the continuation of [loc. cit.].
MSC:
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
37B20 Notions of recurrence and recurrent behavior in dynamical systems
26A18 Iteration of real functions in one variable
37E15 Combinatorial dynamics (types of periodic orbits)
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